Finite Group Databasebeta

← #20276401

Subgroups — #20276401

Order 6784.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811193C211yesmaximal, characteristic
2#101×D53C212yesmaximal, complemented
3#16810421C212yesmaximal, complemented
4#16810420C211yesmaximal, characteristic, complemented
5C53×#355C211yesmaximal, characteristic, complemented
6C2×#11719947C411yescharacteristic
7C2×Q8×D53C2211yescharacteristic
8M16×D53—22no—
9#13133217C411yescharacteristic
10#13133215C2211yescharacteristic
11#13133204C411yescharacteristic
12#13133199—22no—
13#13133194—22no—
14#13133174C411yescharacteristic
15#13133136C2212yes—
16#13133135—22no—
17#124×C53C2211yescharacteristic
18#101×C53C2212yes—
19C2×Q8×C53C2311yescharacteristic
20C2×D4×C53C2×C411yescharacteristic
21C2×C22⋊D53—21no—
22C22×Dic53—21no—
23Q8×D53—42no—
24#11719947—42no—
25C2×C4×D53C2×C411yescharacteristic
26C2×C4×D53D412yes—
27C2×Dic106C2×C411yescharacteristic
28C2×Dic106C2×C411yescharacteristic
29C2×Dic106D412yes—
30M16×C53—22no—
31C22⋊C4×C53—22no—
32C42×C53C2×C411yescharacteristic
33C22⋊Dic53—22no—
34#11719921—21no—
35C4⋊Dic53—22no—
36#11719919—22no—
37#11719919—21no—
38C4×Dic53C2×C411yescharacteristic
39#11719917—22no—
40#11719912—42no—
41C8×D53—42no—
42C23×C53—21no—
43C22×D53C22⋊C411yescharacteristic
44Q8×C53—41no—
45D4×C53—41no—
46C2×C4×C53—21no—
47C2×C4×C53C22×C411yescharacteristic
48C2×C4×C53C2×D412yes—
49C22⋊D53—42no—
50C2×Dic53—21no—
51C2×Dic53—21no—
52C2×Dic53—42no—
53C2×Dic53C22⋊C412yes—
54C2×Dic53C22⋊C411yescharacteristic
55C4×D53—22no—
56C4×D53—22no—
57C4×D53—22no—
58Dic106—41no—
59Dic106—42no—
60Dic106—41no—
61C8×C53—42no—
62#88322—42no—
63C22×C53—42no—
64C22×C53—41no—
65C22×C53C2×C22⋊C411yescharacteristic
66C2×D53—41no—
67C2×D53#9912yes—
68C4×C53—22no—
69C4×C53—21no—
70C4×C53—41no—
71Dic53—21no—
72Dic53—22no—
73Dic53—41no—
74Dic53—41no—
75#2133—531nomaximal, Sylow
76C2×C53—21no—
77C2×C53—81no—
78C2×C53C2×#9911yescharacteristic
79D53—22no—
80#536—531no—
81C2×#101—532no—
82#356—532no—
83#355—531no—
84#355D5311yescharacteristic, complemented
85C53#213311yescharacteristic, complemented, Sylow
86C2×#41—531no—
87C22×Q8—531no—
88C2×M16—1062no—
89C4⋊Q8—531no—
90C4⋊Q8—531no—
91#124—531no—
92#124C2×D5311yescharacteristic
93#123—1062no—
94C22⋊Q8—1062no—
95#117—531no—
96#101—532no—
97#101—1062no—
98#101C2×D5312yes—
99#41—2122no—
100C2×Q8—531no—
101C2×Q8—532no—
102C2×Q8—531no—
103C2×Q8—2122no—
104C2×Q8C22×D5311yescharacteristic
105C2×D4—1061no—
106C2×D4C4×D5311yescharacteristic
107C22×C4—531no—
108C22×C4—532no—
109C22×C4—1061no—
110M16—22no—
111M16—1062no—
112M16—2122no—
113C2×C8—2122no—
114C4⋊C4—1061no—
115C4⋊C4—1062no—
116C4⋊C4—1062no—
117C22⋊C4—22no—
118C22⋊C4—1061no—
119C22⋊C4—1062no—
120C42—531no—
121C42C4×D5311yescharacteristic
122C23—21no—
123C23—531no—
124Q8—41no—
125Q8—2122no—
126Q8—2121no—
127Q8—2121no—
128D4—41no—
129D4—2122no—
130C2×C4—21no—
131C2×C4—532no—
132C2×C4—531no—
133C2×C4—1062no—
134C2×C4—1062no—
135C2×C4—1062no—
136C2×C4—1061no—
137C2×C4—1061no—
138C2×C4—2122no—
139C2×C4C2×C4×D5311yescharacteristic
140C2×C4D4×D5312yes—
141C8—42no—
142C8—2122no—
143C22—42no—
144C22—41no—
145C22—532no—
146C22—2121no—
147C22C22⋊C4×D5311yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—1061no—
152C4—1062no—
153C4—2121no—
154C4—2121no—
155C2—21no—
156C2—81no—
157C2—1062no—
158C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×Q8C22×D53
FittingC53×#355C2
socleC2×C53C2×#99
hypercenter#355D53
hypocenterC53#2133
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #101×D53C2semidirect
3. #16810421C2semidirect
4. #16810420C2semidirect
5. C53×#355C2semidirect
84. #355D53semidirect
85. C53#2133semidirect